#8639769

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#59750	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,9)(4,6)(5,8)(7,10)(11,16)(12,14)(13,15)(17,18,21,22)(19,23,27,25)(20,24,29,30)(26,31,28,32)(33,34)(35,45)(36,42)(37,39)(38,43)(40,44)(41,48)(46,47);(2,4)(3,10)(8,12)(11,13)(17,18,21,22)(19,25,27,23)(20,24,29,30)(26,32,28,31)(33,35,40,48)(34,41,44,45)(36,46,37,43)(38,39,47,42);(2,4)(7,9)(8,12)(15,16)(18,22)(23,25)(24,30)(31,32)(33,36,40,37)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,3,6,10)(2,7,4,9)(5,13,14,11)(8,16,12,15)(17,19)(18,23)(20,28)(21,27)(22,25)(24,32)(26,29)(30,31)(33,37,40,36)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,4,6,2)(3,9,10,7)(5,12,14,8)(11,16,13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,41,37,45);(1,5,6,14)(2,8,4,12)(3,11,10,13)(7,15,9,16)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,39,40,42)(34,37,44,36)(35,47,48,38)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8546965);